Statistics/Distributions/Continuous

A continuous statistic is a random variable that does not have any points at which there is any distinct probability that the variable will be the corresponding number.

General Properties

Cumulative Distribution Function

A continuous random variable, like a discrete random variable, has a cumulative distribution function. Like the one for a discrete random variable, it also increases towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.

Probability Distribution Function

Unlike a discrete random variable, a continuous random variable has a probability density function instead of a probability mass function. The difference is that the former must integrate to 1, while the latter must have a total value of 1. The two are very similar, otherwise. The pdf is represented by a lowercase f.

Special Values

The expected value for a continuous variable is defined as $\int_{-\infty}^{\infty} xf(x)\, dx$

The expected value of any function of a continuous variable g(x) is defined as $\int_{-\infty}^{\infty} g(x)f(x)\, dx$

The mean of a continuous or discrete distribution is defined as E[X]

The variance of a continuous or discrete distribution is defined as E[(X-E[X]$^2$)]

Expectations can also be derived by producing the Moment Generating Function for the distribution in question. This is done by finding the expected value E[e^tX]. Once the Moment Generating Function has been created, each derivative of the function gives a different piece of information about the distribution function.

d¹x$/$d¹y = mean
d²x$/$d²y = variance
d³x$/$d³y = skewness
d⁴x$/$d⁴y = kurtosis